Extreme Value Statistics in Semi-Supervised Models

We consider extreme value analysis in a semi-supervised setting, where we observe, next to the n data on the target variable, n + m data on one or more covariates. This is called the semi-supervised model with n labeled and m unlabeled data. By exploiting the tail dependence between the target variable and the covariates, we derive estimators for the extreme value index and extreme quantiles of the target variable in this setting and establish their asymptotic behavior. Our estimators substantially improve the univariate estimators, based on only the n target variable data, in terms of asymptotic variances whereas the asymptotic biases remain unchanged. A simulation study confirms the substantially improved behavior of both estimators. Finally the estimation method is applied to rainfall data in France. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.